If the vector fields \(u_\varepsilon \)  satisfy an energy upper bound
that is stronger than (REF ),
then we can provide even stronger energy bounds — in particular,
we can show that the energy of \(u_\varepsilon \)  is uniformly bounded
away from a finite set of singularities. These types of results are classical
in the context of Ginzburg-Landau functionals, but we need to
adapt them to our context. Our proof relies crucially on
the analysis in [1]}. We consider
a sequence of vector fields \((u_\varepsilon )_{\varepsilon >0}\)  in \(H^1_{\operatorname{tan}}(M)\) 
that satisfies
\( \begin{split}\omega (u_\varepsilon ) \rightarrow 2\pi \sum _{j=1}^n d_j \delta _{a_j} \qquad \textrm {in } W^{-1,1}(M) \quad \textrm {as } \varepsilon \rightarrow 0,\end{split}\) 
